Optimization of multiple tuned mass dampers to suppress machine tool chatter

ARTICLE IN PRESS International Journal of Machine Tools & Manufacture 50 (2010) 834–842

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International Journal of Machine Tools & Manufacture journal homepage: www.elsevier.com/locate/ijmactool

Optimization of multiple tuned mass dampers to suppress machine tool chatter ˜ oa b, Y. Altintas c,n Y. Yang a, J. Mun a

School of Mechanical Engineering and Automation, Beihang University, Beijing 100191, China Department of Mechanical Engineering, Ideko-Danobat Group, Elgoibar 20870, Spain c Manufacturing Automation Laboratory, Department of Mechanical Engineering, The University of British Columbia, 2054-6250 Applied Science Lane, Vancouver, BC, Canada V6T 1Z4 b

a r t i c l e in f o

a b s t r a c t

Article history: Received 13 August 2009 Received in revised form 9 March 2010 Accepted 22 April 2010 Available online 26 May 2010

Chatter is more detrimental to machining due to its instability than forced vibrations. This paper presents design and optimal tuning of multiple tuned mass dampers (TMDs) to increase chatter resistance of machine tool structures. Chatter free critical depth of cut of a machine is inversely proportional to the negative real part of frequency response function (FRF) at the tool–workpiece interface. Instead of targeting reduction of magnitude, the negative real part of FRF of the machine is reduced by designing single and multiple TMD systems. The TMDs are designed to have equal masses, and their damping and stiffness values are optimized to improve chatter resistance using minimax numerical optimization algorithm. It is shown that multiple TMDs need more accurate tuning of stiffness and natural frequency of each TMD, but are more robust to uncertainties in damping and input dynamic parameters in comparison with single TMD applications. The proposed tuned damper design and optimization strategy is experimentally illustrated to increase chatter free depth of cuts. & 2010 Elsevier Ltd. All rights reserved.

Keywords: Damping Chatter Machine tool

1. Introduction The use of tuned mass damper (TMD) is a classical solution to increase damping of machine tool structures because of its simplicity and lower cost compared with active damping systems [1]. The traditional analytical methods [2] to tune parameters of a single damper are applicable only when the structure is undamped, and considerable efforts have been devoted to address the problem for the case of damped structure with analytical and numerical methods [3–5]. A number of various passive damping systems such as a single TMD with multiple degrees of freedom (MDOF) [6] and multiple TMDs (MTMDs) [7] have been developed to damp a single mode in recent years. It has been demonstrated that multiple TMDs, where each is tuned to damp a specific mode, are more effective than a single TMD solution having the same mass ratio [8]. However, it is difficult to optimize parameters of tuned dampers analytically when the DOFs of TMDs increase. As a result, numerical methods are used, but usually by assuming some restrictions on the design of stiffness or damping ratio of each TMD [8]. Recently, Zuo and Nayfeh [9] proposed to optimize individual stiffness and damping elements of multiple TMDs by applying the principles of a decentralized H2 control method in

n

Corresponding author. E-mail address: [email protected] (Y. Altintas).

0890-6955/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmachtools.2010.04.011

order to minimize the root mean squared (RMS) response of the main structure. Li and Ni [10] were able to optimize the MTMDs numerically in order to minimize vibration amplitude without imposing restrictions on design parameters of the TMDs. These methods aimed to minimize RMS value of frequency response when excitation force is random [8,9] or when excitation force is harmonic [10]. However, the tuning requirements of chatter suppression are quite different in machining. Chatter is a self-excited vibration problem that causes poor surface finish and large dynamic loads on the spindle. Tobias and Fishwick [11] and Tlusty and Polacek [12] related absolute stability of the cutting process to the minimum real part of tool point frequency response function (FRF). Although stability of metal cutting is a complex problem, there are many cases where stability does not depend directly on oriented real part of FRF [1]. However, stability of general machining operations like turning, boring, or milling with one dominant mode can be predicted reasonably using Tlusty’s method [12] by orienting real part of FRF in the direction of chip regeneration [1]. Some experimental studies have been reported about the application of TMDs on machine tools for chatter suppression. Rivin and Kang [13] used a tuned damper to improve available length-to-diameter ratio in boring. Tarng et al. [14] and Rashid and Nicolescu [15] tuned damper’s natural frequency to match with natural frequency of the structure’s target mode, which

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needs to be damped. However, optimal tuning strategies to suppress chatter by considering stability of metal cutting systems have not been studied except by Sims [16]. Sims tuned a single TMD to push the negative real part of main structure’s FRF analytically. Saffury and Altus [17] also tuned parameters of a viscoelastic cantilever beam by maximizing the most negative real part of FRF. However, the listed methods have considered the tuning of only a single TMD to reduce chatter. This paper presents optimal tuning of both single and multiple TMDs to suppress chatter in machining. The experimentally validated optimization strategy allows evaluation of multiple dampers on chatter stability without increasing damper’s mass. Henceforth, the paper is organized as follows. The chatter stability law is turned into an objective function to reduce unstable vibrations in machining in Section 2. Optimal identifications of frequency and damping ratios of TMDs are presented in Section 3. Sensitivity of the objective function, i.e. chatter stability law, to tuning parameters is discussed in Section 4. The performances of single and multiple TMDs in reducing chatter are presented in Section 5. The paper is concluded in Section 6.

only the case of positive a will be discussed here. While the objective is to reduce magnitude of FRF in forced vibration cases, the focus is to shrink the negative real part of FRF in chatter avoidance as proposed by Sims [16].

3. Optimal identification of TMD parameters The objective of multiple TMDs is to damp the dominant structural mode described by its mass (m0), stiffness (k0), and damping constant (c0). A mechanical system with multiple TMDs attached to the structure that needs to be damped is shown in Fig. 2. Each TMD has a single degree of freedom (DOF) with mass mj connected to the main structure with a spring kj and a damper cj (j¼1, y, N). The equations of motion for main mass (m0) and TMD mass (mj) in xj direction are 9 N X > = m0 x€ 0 þ c0 x_ 0 þ k0 x0 þ ½cj ðx_ 0 x_ j Þ þ kj ðx0 xj Þ ¼ F sin ot > ð2Þ j¼1 > > ; _ _ € m x c ðx x Þk ðx x Þ ¼ 0 j j

2. Chatter stability objective The regenerative effect of chatter can be explained as in Fig. 1. Chip thickness starts deviating dynamically from its static value (h0), i.e. feed per revolution in orthogonal cutting, when the system experiences vibrations. Dynamic chip thickness, which is the root cause of regenerative chatter, oscillates due to vibration marks left during the previous cuts y(t  T) and current cuts y(t), which in turn causes cutting force to oscillate. Depending on phase shift e between the inner y(t  T) and outer y(t) waves, dynamic chip thickness (h(t)) may grow exponentially and cause the cutting system to become unstable and experience chatter. Stability of this system depends not only on dynamic properties like mass, damping, and stiffness but also delay between vibrations, which is important as well. Mathematically, this fact drives real part dependent stability [11,12,18]. Stability charts that show critically stable depth of cut (alim) at each spindle speed (n) can be evaluated using the following stability law proposed by Tobias and Fishwick [11] and Tlusty and Polacek [12]: 9 1 > alim ¼ > 2aKf GðoÞ =   ð1Þ o > > ; n ¼ 60 2kp þ e where a is the directional factor, Kf the cutting force coefficient, G(o) the real part of relative FRF between tool and workpiece, and o the chatter frequency. Depending on the sign of a, critical depth of cut is determined by the minimum negative real part (G(o)) of FRF or the maximum positive real part. Without loss of generality,

835

j

0

j

j

0

j

where j¼1, y, N is the TMD index and F sin(ot) is the harmonic excitation force acting on the main structure. Transforming the equations of motion into Laplace domain, the transfer function between the vibration of main mass and external force can be expressed as

F0 ðsÞ ¼

X0 ¼ F

1 m0 s2 þc0 s þ k0 þ

N P

:

j¼1

Introducing non-dimensional terms as listed in Table 1 and switching to frequency domain (s¼io), the frequency response function can be represented as a function of excitation frequency ratio (b ¼ o/o0), mass ratio (mj ¼mj/m0), damping ratio (xj), and natural frequency ratio (fj ¼ oj/o0) of tuned dampers:

F0 ðbÞ ¼

1 1 , k0 gðbÞ þ hðbÞUi

ð4Þ

where 2

gðbÞ ¼ 1b b

n X

2

2

mj

j¼1

hðbÞ ¼ 2bx0 þ b

2

n X j¼1

mj

2

fj2 =b 1 þ 4xj

2

,

2

:

ðfj =bb=fj Þ2 þ 4xj 2xj b=fj ðfj =bb=fj Þ2 þ 4xj

The real part of F0(b) can be expressed as G0 ðbÞ ¼

1 gðbÞ : k0 gðbÞ2 þ hðbÞ2

ð5Þ

The objective of chatter suppression is to maximize the minimum of G0(b). All TMDs are designed with identical mass, and mass ratio m is usually given prior to designing a TMD mj

m1 k1

c1

kj

mN cj

kN

cN

xj

m0

x0 k0

Fig. 1. Regenerative chatter effect in turning.

ð3Þ

ðcj s þ kj Þmj s2 =mj s2 þ cj sþ kj

F sin ω t

c0

Fig. 2. SDOF structure with multiple TMDs.

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Table 1 Definition of non-dimensional parameters. TMD (j¼1, y, N)

Main structure

Mass ratio

Damping ratio

mj ¼ mj/m0

xj ¼ cj =2 kj mj

qffiffiffiffiffiffiffiffiffiffi

Frequency ratio

Natural frequency

fj ¼ oj/o0

oj ¼

qffiffiffiffiffiffiffiffiffiffiffiffi kj =mj

Frequency ratio

Damping ratio

b ¼ o/o0

x0 ¼ c0 =2 k0 m0

pffiffiffiffiffiffiffiffiffiffiffiffi

Natural frequency

o0 ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi k0 =m0

Fig. 3. Comparison of objective function (J) among real part based, magnitude based (HN), and H2 tuning for a single TMD system with m ¼ 5%. (a) Damping ¼0%, (b) damping¼ 10%.

Fig. 5. Minimum of optimum J using the real part based tuning for different x0 (x0 ¼ 0%, 2%, 5%, 10%).

Fig. 4. Plot of J as a function of fj and xj with m ¼ 5% and x0 ¼0 (N ¼ 1).

because of space restriction for added mass on the machine. The parameters that remain to be optimized are frequency (i.e. stiffness kj of each TMD since m1 ¼ m2 ¼    ¼ mj ) and damping ratios of each TMD (i.e. x1, y, xj, y, xn, j ¼1, 2, y, N). The objective function J can be defined proportional to G0(b), Jðb,VÞ ¼

gðbÞ gðbÞ2 þ hðbÞ2

,

where, V ¼{f1, x1,y, fj, xj,y, fN, xN }, j ¼1, 2, y, N.

ð6Þ

The task of optimization is to shrink the negative real part of FRF for chatter resistance, i.e. maximize the minimum of the objective function J. The minimax numerical method as proposed by Zuo and Nayfeh [19] is employed to identify stiffness and damping ratio of each TMD through maximizing the minimum real part instead of minimizing the magnitude of FRF for chatter avoidance. Details of the minimax method can be found in [20], and only its brief application to tuning of dampers is given as follows.

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(2) Search direction: discretize J (Eq. (6)) by varying excitation frequency ratio b around resonance, as the TMDs influence J only in this narrow zone. Identify dl at which J(dl, Vl) are close to the minimum of J within a given tolerance e:

The following algorithm of minimax pushes J towards zero (J-0), identifies the optimal frequency ratios and damping ratios of N TMDs. Hence there are 2N variables. Objective: max{min(J(b, V))} (i.e. push the negative real part of FRF towards zero) Design variables: V¼{f1, x1, y, fj, xj, y, fN, xN}, j ¼1, 2, y, N

Ie ¼ fl9ðJðdl ,Vl ÞminJðb,Vl ÞÞ r e, l A Ig: Compute gradient rJ with respect to xj and fj for all points in Ie and create a convex hull conv(rJ). Solve a constrained linear least square problem to obtain the search direction

(1) Choose initial values V0 ¼{(fj)0, (xj)0}, j¼ 1, 2,yN, which are selected to be the same for all TMDs. The initial values can be selected by tuning stiffness and damping of a single TMD to achieve the maximum real part of FRF as suggested by Sims [16].

S ¼ arg min:d:,

d A convðrJÞ:

Table 2 The effect of number of TMDs on objective function J with m ¼5% for an undamped main structure (x0 ¼ 0). Number of TMDs N

1

2

3

5

10

30

50

Minimum J Improvement against single TMD

 2.703 —

 2.348 13%

 2.202 19%

 2.073 23%

 1.956 28%

 1.861 31%

 1.836 32%

Fig. 6. Robustness analysis of the multiple TMD system for m ¼5% and x0 ¼0. Uncertainties are assumed to be independent and to satisfy a normal distribution. (a) Effect of the TMDs stiffness uncertainties (mean value 0, standard deviation 15%), (b) effect of the TMDs stiffness uncertainties (mean value 0, standard deviation 7.5%), (c) effect of the TMDs damping uncertainties (mean value 0, standard deviation 15%) and (d) effect of the input parameter uncertainties (mean value 0, standard deviation 15%).

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For an arbitrary set GCI, the convex hull can be defined as [20] 8 r < X convðGÞ ¼ X ¼ aj Xj 9Xj A G; :

aj Z 0,

j¼1

r X j¼1

9 = aj ¼ 1,r ¼ 1,2,. . . : ;

(3) If 99S99 is small enough, terminate the optimization algorithm. Otherwise continue with the next step. (4) Increase the variables by g, and solve a one-dimensional optimization to determine the optimum g that maximizes the function minJðd,Vl þ gSÞ: (5) Let Vl + 1 ¼Vl + gS, l ¼l +1, go to (2).

4. Non-dimensional sensitivity analysis of TMD parameters There are several design principles that improve damping of a flexible mode through tuned mass dampers. The traditional approach is to reduce magnitude of FRF at resonance frequency, but the minimum real part of the mode is targeted here to reduce chatter. The most optimal chatter resistance corresponds to

having an objective function approaching zero (J-0), which indicates infinite rigidity. 4.1. Single TMD design results Using the non-dimensional parameters listed in Table 1, performance of tuning by targeting real part of FRF against its magnitude (HN) and H2 is compared in Fig. 3. The objective function is 34% and 27% larger in the real part based optimization in comparison with magnitude based tuning (x0 ¼0% and x0 ¼10%), which yields higher chatter free axial depth of cut as indicated by Eq. (1). The sensitivities of TMD performance to additional design parameters are also investigated using non-dimensional analysis. The sensitivities of frequency and damping ratios of the TMD on objective function are shown in Fig. 4. The tuned damper can be designed to have damping ratio between 2% and 20% in practice. Natural frequency of the tuned damper is never equal to the mode’s resonance frequency, but it is always greater in order to shrink real part of FRF. This is achieved by creating positive real part for TMD at a frequency where the structure’s real part is negative minimum. However, when multiple TMDs are used, natural frequencies of the TMDs may spread above and below the

Fig. 7. Experimental set-up and structural mode to be damped (o0 ¼ 315 Hz, x0 ¼2.76%, k0 ¼ 59.5 N/mm, m0 ¼ 15.2 kg) ( fixture mounted on a lathe turret, (b) measured FRF at the tool tip in feed direction and (c) mode shape of 315 Hz.

undeformed shape). (a) Experimental

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Fig. 8. Experimental comparison of a single TMD and three TMDs applied: (a) magnitude of the FRF and (b) real part of the FRF.

natural mode of the structure. Optimal values of damping ratio and frequency ratio (i.e. stiffness) of a single TMD are coupled to achieve optimum objective function. Simulations indicate that increased mass of the TMD does not affect objective function proportionally, and a typical mass ratio range m E1–5% is sufficient regardless of main structure’s damping ratio (x0), see Fig. 5. Furthermore, adding larger mass to the machine tool structure is not desirable in practice. 4.2. Effect of number of dampers It is assumed that all tuned dampers have the same mass, but tunable stiffness and damping values. The effect of number of TMDs is shown in Table 2 for a sample undamped structure. Total mass of the tuned dampers is set to 5% of structure’s modal mass. It is shown that objective function improves as N increases, but it is not significantly affected after adding three or more TMDs. Therefore, selection of the best number of TMDs involves choosing a good balance between cost and gained chatter resistance while keeping added total mass always the same.

Table 3 Optimum fj and xj of three TMDs for damping 315 Hz mode (j¼ 1, 2, 3).

fj

xj

TMD 1

TMD 2

TMD 3

0.987 6.29%

1.077 6.76%

1.184 7.24%

Table 4 Cutting conditions for the chatter stability simulation. Insert

Workpiece

Material Nose radius Inclination angle Rake angle

Carbide 0.0315 mm 51 51

Cutting edge angle

951

Material Diameter Overhang Tangential cutting force coefficient Radial cutting force coefficient

Steel AISI 1045 41 mm 90 mm 2766 MPa 1544 MPa

4.3. Robustness of the TMD design Since it is impossible to manufacture a TMD to match the design exactly, robustness of damping is simulated with independent parameter uncertainties on all TMDs and input parameters. The uncertainties are assumed to satisfy a normal distribution in order to make the simulation more realistic. As tuning of stiffness and damping could be implemented independently, robustnesses against TMD stiffness error, TMD damping ratio error, and input target mode parameter error are analyzed separately; 10,000 simulations are run for each case (N¼1, 2, 5, 10), and robustness curves are represented by cumulative distribution functions, which denote probability of the event that J is larger than or equal to the value on X-axis. It is found that J is very sensitive to stiffness variations when TMDs are employed (Fig. 6(a) and (b)), while J is less sensitive to uncertainties in damping ratio (Fig. 6(c)). Robustness against stiffness variations starts to reduce fast after N¼5. Fig. 6(d) shows that multiple dampers are more robust to changes in input dynamic parameters. In short, multiple TMDs need more accurate tuning of stiffness and natural frequency of each TMD, and the system is more robust to changes in damping and input dynamic parameters after fulfilling this condition.

Fig. 9. Stability chart comparison for different tunings.

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5. Experimental verification Performances of single and multiple TMDs are verified on a specially designed fixture of a CNC turning machine (Fig. 7). Measured FRF at the tool tip exhibits three distinct modes at 107 and 233 Hz, which originate from the machine tool structure, and 315 Hz contributed by the fixture. Unlike the 107 Hz suspension

mode, the 315 Hz mode with 15.2 kg modal mass contributes to relative vibrations between the workpiece and tool in the feed (i.e. regenerative chip) direction; hence it is selected as a target mode to be damped. Fig. 8 Three identical, single DOF TMDs are designed with a total mass ratio of 4.2%. The TMD contains a high density mass block cantilevered to the fixture with a slender beam. The mass can be

Fig. 10. Vibrations of the tool in the feed rate direction for the cutting condition a¼ 1 mm, n ¼2000 rpm, and f ¼0.1 mm/rev: (a) cutting test without TMD, (b) cutting test with a single TMD and (c) cutting test with three TMDs.

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moved along the beam to adjust stiffness, and damping is varied by changing screw resistance at the mass–cantilever beam connection. The TMDs are placed at the fixture and along the direction of vibration, where the mode has the highest amplitude.

841

real part of the target mode is increased from G0 ¼ 0.181 mm/N to G0 ¼  0.041 mm/N, which corresponds to 4.4 fold increase in chatter free depth of cuts to the target mode of 315 Hz. 5.2. Chatter stability tests

5.1. Effect of TMD parameters A single TMD is designed with a mass ratio of 1.4% (m1 ¼0.213 kg). The damping and frequency ratios are identified as x1 ¼7.80% and f1 ¼1.051 (i.e. o1 ¼1.051  315E331 Hz) by applying the minimax optimization procedure. The magnitude of FRF is reduced from 0.313 to 0.156 mm/N, which corresponds to doubling of dynamic stiffness and reduction of the forced vibrations. The negative peak of real part of FRF is increased from G0 ¼  0.181 mm/N to G0 ¼ 0.077 mm/N, which corresponds to 2.4 fold increase in chatter free axial depth of cuts. Next, three TMDs are designed, each having a mass ratio of 1.4% (mj ¼0.213 kg) and a total added mass of 4.2% (0.639 kg). The optimum frequency (fj) and damping (xj) ratios for each TMD have been identified as listed in Table 3. Magnitude of the target mode is further reduced from 0.313 to 0.098 mm/N, and the minimum

Chatter stability is predicted in turning AISI 1045 steel with a tool and cutting force coefficients as listed in Table 4. Stability lobes consider only the 315 Hz mode in order to illustrate the effect of TMDs on chatter resistance. Four stability lobes are plotted by considering undamped fixture, damped with single TMD of m ¼1.4%, damped with single TMD of m ¼4.2% and three TMDs in Fig. 9, where the case of single TMD with m ¼4.2% is simulated. The minimum depth of cut is increased from 1.0 to 2.5 mm with a single TMD of m ¼1.4%, 3.9 mm with a single TMD of m ¼4.2%, and 4.6 mm can be reached when three TMDs are used. Series of turning tests have been conducted to validate the performance of TMDs on chatter stability. The overhang of the AISI 1045 workpiece bar is kept short in order to keep it significantly rigid against the modes at the tool tip. In order to

Fig. 11. Vibrations of the tool in the feed rate direction for the cutting condition a ¼4 mm, n¼ 2000 rpm, and f ¼0.1 mm/rev: (a) cutting test with a single TMD and (b) cutting test with three TMDs.

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avoid exciting the 107 and 233 Hz modes, which became more flexible after damping of the 315 Hz mode by TMDs, cutting conditions are selected to excite the 315 Hz mode only. Vibrations are measured with an accelerometer attached to the fixture. Cutting test results conducted at a¼1 mm depth of cut, n ¼2000 rpm spindle speed, and f¼0.1 mm/rev feed rate are shown in Fig. 10. The system chattered severely when the fixture was not damped at 326 Hz with a vibration amplitude of 2.0g. The chatter disappeared when turning under the same cutting condition with a single TMD and three TMDs. Vibration amplitude was reduced from 2.0g to 0.14g with a single TMD. Vibration amplitude was reduced to almost air cutting (0.1g) when three TMDs were used. Additional cutting tests have been conducted by increasing depth of cut from a¼1 mm to a¼4 mm at the same spindle speed (n¼2000 rpm) and feed rate (f¼ 0.1 mm/rev)). The single TMD case experienced chatter, whereas the fixture damped with three TMDs was chatter free as shown in Fig. 11. Other dominant modes of the machine, which became more flexible after damping the fixture mode, can also be damped with the proposed damping strategy.

6. Conclusions Machine tools are generally damped using mass dampers tuned to natural frequency of dominant modes aligned with the principal direction of cutting. Such a strategy reduces magnitude of FRFs at natural frequency, which also reduces forced vibrations. However, chatter stability is most sensitive to the minimum negative real part of FRF, which is located above natural frequency of the machine. This paper confirms that chatter free axial depth of cuts can be improved by several folds using mass dampers tuned to maximize negative real part of FRFs as suggested by Sims [16]. It is shown that further significant gains can be achieved using more but identical TMDs without increasing total mass of the dampers noticeably. Optimum identification of TMD stiffness and damping parameters have been carried out with a numerical minimax method, which is more robust to initial conditions than commonly used gradient methods. The proposed damping strategy can be extended to milling operations by considering the chatter stability law of periodic systems [21].

Acknowledgements The tuned dampers were designed by Dr. Wang Min from Beijing University of Technology. This research is supported by NSERC Discovery Grant.

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